Affine Phase Retrieval for Sparse Signals via ℓ1 Minimization

Affine phase retrieval is the problem of recovering signals from the magnitude-only measurements with a priori information. In this paper, we use the ℓ 1 minimization to exploit the sparsity of signals for affine phase retrieval, showing that O ( k log ( e n / k ) ) Gaussian random measurements are sufficient to recover all k -sparse signals by solving a natural ℓ 1 minimization program, where n is the dimension of signals. For the case where measurements are corrupted by noises, the reconstruction error bounds are given for both real-valued and complex-valued signals. Our results demonstrate that the natural ℓ 1 minimization program for affine phase retrieval is stable.